A STUDY OF THE EFFECT OF MOBILITY RATIOS ON PATTERN DISPLACEMENT BEHAVIOR AND STREAMLINES TO INFER PERMEABILITY FIELDS PERMEABILITY MEDIA SUPRI TR 115 By Yuandong Wang Anthony R. Kovscek William E. Brigham December 1998 Work Performed Under Contract No. DE-FG22-96BC14994 Prepared for U.S. Department of Energy Assistant Secretary for Fossil Energy Thomas Reid, Project Manager National Petroleum Technology Office P.O. Box 3628
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A STUDY OF THE EFFECT OF MOBILITY RATIOS
ON PATTERN DISPLACEMENT BEHAVIOR AND
STREAMLINES TO INFER PERMEABILITY
FIELDS PERMEABILITY MEDIA
SUPRI TR 115
By
Yuandong Wang
Anthony R. Kovscek
William E. Brigham
December 1998
Work Performed Under Contract No. DE-FG22-96BC14994
Prepared forU.S. Department of Energy
Assistant Secretary for Fossil Energy
Thomas Reid, Project ManagerNational Petroleum Technology Office
P.O. Box 3628
ii
TABLE OF CONTENTS
List of Figures iii
Acknowledgments iv
Abstract v
Chapter 1 Effect of Mobility Ratio on Pattern Behavior of
A Homogeneous Porous Medium ..................................................... 1
1.1 Introduction .......................................................................................... 1
1.2 Sweep Efficiency ................................................................................. 3
1.3 Simulation Results ............................................................................... 4
1.4 Breakthrough Determination ............................................................... 7
1.5 Discussion of Pattern Behavior ........................................................... 9
1.5.1 Areal Sweep Efficiency ......................................................... 10
1.5.2 Unit Mobility Ratio ............................................................... 13
1.5.3 Five-Spot Pattern, Very Favorable Mobility Ratio ............... 15
1.5.4 Staggered Line Drive, Very Favorable Mobility Ratio ......... 17
1.6 Conclusions ......................................................................................… 19
Appendix A—Effect of Grid-block and Time-step Size .........................……….. 21
Appendix B—Effect of Grid-block Orientation .....................................………... 24
Appendix C—Numerical Dispersion ......................................................……….. 27
Chapter 2 Streamlines to Solve Inverse Problems .......................................... 29
2.1 Definition of the Research Topic ....................................................... 29
2.2 Importance of the Research ............................................................... 35
2.3 Future Work Plan and Expected Results ........................................... 35
Nomenclature 37
References 38
iii
LIST OF FIGURES
Chapter 1Fig. 1 Dyes et al (1954) Experimental Results ........................................ 2
Fig. 2 Comparison of Simulation Results and
Experimental Results by Dyes et al .....................................…… 6
Fig. 3 Fractional Flow(fw) vs tD at Producer—Determination of
Breakthrough Time ...................................……………………… 8
Fig. 4 Effect of Mobility Ratios and Pattern Geometry
on Areal Sweep Efficiency .....................................….................. 11
Fig. 5 Streamline and Saturation Distribution Unit Mobility Ratio
...............................................................……………………. 14
Fig. 6 Five-Spot Pattern, Very Favorable
Mobility Ratio, 1/M = 20 ...................................................……... 16
Fig. 7 Staggered-Line-Drive Pattern, Very Favorable Mobility Ratio,
1/M =20 ...........................………………………………………. 18
Fig. A Effect of Refining Grid and Increasing Time steps on the
simulation Accuracy ..............................………………..........…. 22
Fig. B.1 Grid Orientation………………………………………………. 24
Fig. B2 Effect of Grid Orientation on Simulation Results .............……. 26
Fig. C Displacing Front ..................................................................……. 28
Chapter 2Fig. 1 Comparison of Permeability Field ......................................……... 31
Fig. 2 Comparison on Breakthrough Curve ....................................…….. 32
Fig. 3 Error vs Iterations ..................................................................……. 34
iv
ACKNOWLEDGMENTS
This work was supported by the Assistant Secretary for Fossil Energy, Office of
Oil, Gas, and Shale Technologies of the U.S. Department of Energy under contract No.
DE-FG22-96BC14994 to Stanford University. Likewise, the support of the SUPRI-A
Industrial Affiliates is gratefully acknowledged.
v
ABSTRACT
In my MS research program I worked on two research topics. I finished the first
research topic and conducted some preliminary research on the second topic. The second
topic is actually a first step toward my Ph.D. program. Below is a brief description of
my work on the two topics.
1. A study of the effect of mobility ratios on pattern displacement behavior
It is well known, for unit mobility ratio, that the areal sweep efficiency of a
staggered line drive pattern is always better than a five-spot pattern. However, this
observation does not hold for very favorable mobility ratios.
I studied the effect of mobility ratios on pattern behavior by the means of
simulation using a streamline simulator. In this report, I present simulation results and,
with the help of streamline and saturation distributions, explain the differences between
displacements with unit and favorable mobility ratios. Simulations compare well with
experiments conducted elsewhere.
Accurate definition of breakthrough time is also discussed for multiphase,
streamline, simulation results. The exact definition of breakthrough is difficult due to
physical dispersion in experiments and numerical dispersion in simulations.
2. Streamline approach to the inverse problem of inferring permeability distribution
It is often useful to infer the permeability field of a porous medium, such as an oil
reservoir or a ground water aquifer from a tracer breakthrough curve at the production
well. There are many approaches to this inverse problem, such as simulated annealing,
sensitivity studies either analytical or numerical, genetic algorithms, and geo-statistical
approaches. Most of these approaches are very time consuming.
vi
An efficient approach is desirable. Here I propose an alternative approach—an
inverse streamline approach. I have conducted some preliminary work on this topic, to
be described later, and obtained some satisfactory results. These results indicate that this
project is robust and promising. However, there are simplifications in my current study,
such as piston-like displacement, unit mobility ratio and incompressible flow. If these
simplifications can be relaxed, then this can be a very efficient approach to the inverse
problem with broader applications.
1. EFFECT OF MOBILITY RATIO ON PATTERN BEHAVIOR
OF A HOMOGENEOUS POROUS MEDIUM
1.1 INTRODUCTION
Pattern geometry plays a major role in determining oil recovery during secondary
and enhanced oil recovery operations. Although simulation is an important tool for
design and evaluation, the first step often involves rough calculations based upon areal
sweep efficiencies of displacements in homogeneous, two-dimensional, scaled, physical
models (Dyes et al., 1954; Craig, 1971; Lake, 1989). These results are available as a
function of the displacement pattern and the mobility ratio, M. The mobility ratio is
simply the mobility of the displacing phase over that of the displaced, or resident, phase.
Because it is possible to compute sweep efficiency analytically when the displacing and
displaced phase have the same mobility (Morel-Seytoux, 1966), scaled physical model
results have been verified for unit mobility ratios.
Convincing verification of the non-unit mobility ratio cases does not appear in the
literature. Typical finite difference solutions of the reservoir flow equations suffer from
numerical dispersion, the effects of which are hard to evaluate. Furthermore, the scaled
physical model results at low mobility ratios (M << 1) are provocative. For instance,
Fig. 1a shows that recovery from a five-spot pattern at breakthrough for 1/M greater than
about 6 is virtually 100%, whereas recovery at breakthrough in Fig. 1b for a staggered-
line-drive pattern at an 1/M of 6 is only about 88.5%. This contradicts the common
notion that areal sweep efficiency from a staggered-line-drive pattern is always better
than that from a five-spot pattern.
We use a 3D streamline simulator (Batycky, et al., 1996) to analyze
displacements in five-spot and staggered-line-drive patterns for stable displacements, that
is M less than 1. In the following sections, we present streamline distributions, saturation
distributions, and fractional flow at the producer versus dimensionless time, tD. The
dimensionless time is the pore volumes of displacing fluid injected. With the streamline
2
Fig 1 Dyes et al(1954) Experimental Results
16
(a) Five Spot Pattern
(b) Staggered Line Drive, d/a = 1
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and saturation distributions at different times, we explain why and at what mobility ratio
the five spot pattern can recover more oil than a staggered line drive pattern.
The streamline calculation method is advantageous in that the results suffer from
much less numerical dispersion than typical finite-difference approximations, but some
dispersion in simulation results is evident. Therefore, we discuss how to treat the
numerical dispersion to obtain accurate estimates of breakthrough times. We discuss the
proper way to calculate fractional flow based on the flow rates at the producer. In
comparing the simulation results with the experimental results of Dyes et al. (1954),
physical dispersion in the experiments is found even though a piston-like displacement
was assumed.
1.2 SWEEP EFFICIENCY
Before proceeding, it is useful to recall the representation of experimental data in
Fig. 1 and the meaning of sweep efficiency. Dyes et al. (1954) used various oils as both
the injected and displaced phases. These hydrocarbons were miscible and they assumed
piston-like displacement. An X-ray shadowgraph technique was used to observe the
position of the displacing front. Areal sweep efficiencies are plotted versus displacable
pore volumes injected for different mobility ratios. In the figure, the x axis is the
reciprocal of mobility ratio. Each curve in the graphs corresponds to a specific tD / ∆ S ,
or displacable pore volume injected. The bottom curves show sweep efficiencies at
breakthrough. It is assumed that the displacement has a piston-like front and there is no
physical dispersion. Likewise, the porous medium is assumed to be perfectly
homogeneous.
For piston-like displacement, the areal sweep efficiency is
EA = AS / AT (1)
4
where As is the swept area and AT is the total area. Before and at breakthrough, the
amount of displacing fluid injected is equal to the displaced fluid produced, disregarding
compressibility. Assuming piston-like displacement, injected volume is related to area
swept
VI = AS hφ∆ S (2)
where VI is the volume of displacing fluid injected, h is the thickness of the formation,
and φ is porosity. Hence,
EA = AS / AT = VI / (AT hφ ∆ S) = tD / ∆ S (3)
where tD = V I / (AT hφ) is the pore volume of fluid injected, also commonly called
dimensionless time. For ∆ S =1, EA = tD before and at breakthrough.
After breakthrough,
EA = (VI − VP ) / ( AT hφ ∆ S) (4)
where VP is volume of displacing fluid produced.
1.3 SIMULATION RESULTS
We use a three-dimensional streamline simulator, called 3DSL, written by
Batycky et al (1997) to simulate the displacement for the five spot and staggered line
drive patterns. In the simulations, we set the conditions identical to those in the
experiments and choose relative permeability curves that ensure a piston-like
displacement. The conditions are
1. Homogeneous permeability field, i.e., k is constant.
5
2. Straight line relative permeability curves with end point relative permeability of
1, i.e.,
krw = Sw, kro = Sw (5)
Therefore, krw + kro = 1 for any Sw .
4. Mobility is altered by changing viscosity, and the mobility ratio is the
reciprocal of viscosity ratio.
5. We set ∆ S =1 which means that ahead of the displacing front, the displacing
phase saturation is zero, and behind the front, it is unity.
3DSL is very fast compared to conventional finite difference simulators and
exhibits much less numerical dispersion (Batycky et al., 1997; Thiele, et al., 1996). For
our problem, it offers us the streamline distribution which facilitates explanation of
displacement behavior.
I use many pressure solves (time steps) and very fine grids (100 by 100 cells for
the five spot and 140 by 70 cells for the staggered line drive) to ensure converged
simulation results. A grid refinement study (refer to Appendix A for details) showed
these grids to be optimal in that further refinement of the grid did not yield noticeable
changes in breakthrough time, the oil production curve, or displacement patterns. For unit
mobility ratio (M = 1), we actually only need one pressure solve. But for mobility ratios
far from 1, we need many pressure solves. For 1/M = 20, we used up to 1000 pressure
solves to ensure that the results were converged. In the streamline approach, a pressure
solve is accompanied by a re-determination of streamline paths; hence, the flow field.
I also studied the effect of grid orientation on the simulation results (see Appendix
B) and found that the pattern behavior is independent of the grid orientation.
Figure 2 displays areal sweep efficiencies as a function of pore volume injected
for differing mobility ratios, and compares simulation and experimental results (Dyes et
al., 1954). The solid lines are simulation results, solid circles experimental results, and
dotted lines connect circles for ease of viewing. In this figure, we concentrate on only the
favorable mobility ratios, M < 1. We noticed more numerical dispersion for unfavorable
mobility ratio cases, not reported here.
6
Fig 2 Comparison of Simulation and Experimental
17
(a) Five Spot Pattern
(b) Staggered Line Drive, d/a = 1
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Figure 3 shows the displacing fluid fractional flow at the producer as a function of
dimensionless time for several mobility ratios. To compute fractional flow from the
numerical data we use a central finite-difference formula rather than backward
differences. The shapes of the fractional flow curves at breakthrough (tD from roughly
0.7 to 1) indicate some numerical dispersion. We expect the fractional flow to increase
sharply rather than gradually at water breakthrough. As expected, the numerical
dispersion decreases as M becomes more favorable (Peaceman, 1977). The most
numerical dispersion occurs for unit mobility ratio, as shown in Fig. 3. We modify the
breakthrough time by trimming the numerical dispersion as will be described next.
1.4 BREAKTHROUGH DETERMINATION
Due to numerical dispersion, injected fluid breaks through earlier at the producer
than it should. However, the numerical dispersion does not have much effect on the late-
time displacing fluid production. The fractional flow versus tD plots shown in Fig. 3
illustrate the early breakthrough caused by numerical dispersion.
To correct for numerical dispersion in breakthrough times and approximate the
breakthrough time more accurately, we use fractional flow data after breakthrough and
extrapolate back to breakthrough time. A least-squares method is used with second order
polynomials:
tD = a + bfw + cfw2 (6)
The data points employed lie between 0.1 < f w < 0.5. The dashed lines in Fig. 3 illustrate
this procedure. All the breakthrough times in Fig 2 are modified using this method.
Numerical dispersion is also related to the number of time steps (i.e., pressure
solves). In the streamline approach applied here, dispersion can be introduced through the
process of mapping the streamline saturation distribution onto the underlying Cartesian
grid used to compute the pressure field5. Hence, for a unit mobility ratio where the
8
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Fig. 3 Fractional Flow(fw) vs tD at Producer—Determination of Breakthrough Time
9
pressure field does not change, the most accurate results are obtained when a single time
step is used. By performing various single time step simulations, we determine a
breakthrough time of 0.7178 for the five spot with M equal to 1. This is in good
agreement with the analytical solution (Morel-Seytoux, 1966) of 0.7177. Likewise,
Morel-Seytoux analytically and numerically determined breakthrough times for a
staggered-line-drive pattern are both equal to 0.785. With multiple time steps (50)
breakthrough times were sooner, but using the method discussed above, we obtain the
same values for the breakthrough times, at M = 1.
1.5 DISCUSSION OF PATTERN BEHAVIOR
Comparing simulation results and experimental results in Fig. 2, we notice that
breakthrough occurs earlier in the experiments than predicted by the simulations. For the
staggered-line-drive pattern in Fig 2(b), experimental data indicates that EA at
breakthrough is roughly 0.75 for M equal to 1. In the experiments, there is physical
dispersion even though a piston-like front is assumed. In the simulation results presented
in Fig. 2a, the sweep efficiency at breakthrough is 99.7% when 1/M = 20. An almost
negligible area immediately around the producer is not completely swept at breakthrough
due to a very small amount of dispersion in the simulations (refer to Fig. 3, five spot, 1/M
= 20).
After breakthrough, the differences in areal sweep efficiencies between the
experiments and simulations become much smaller (Fig. 2). After breakthrough, the
numerical dispersion consists of only a portion of the displacing fluid produced. As time
increases, this portion decreases and the dispersion has less effect on areal sweep
efficiency. However, the differences between the experimental results and the simulation
results are consistent, i.e. the areal sweep efficiencies of the simulations are generally
higher than those of the experiments. As shown in Fig. 2(a), the simulated breakthrough
curve levels off at large 1 / M with zero slope and does not reach 1 at 1/M = 20. A plot
of displacing front position at late displacement time shows that a very small amount of
numerical dispersion causes slightly earlier breakthrough. Further details are given in
10
Appendix C. However, the experimental curve shows EA equal to 1 at 1 / M equal to
about 7.5. We note that in the plot drawn by Dyes et al, the point where the breakthrough
curve hits the EA equal to 1 line is only an extrapolation from other data points.
1.5.1 Areal Sweep Efficiency
Figure 4 plots computed breakthrough, tD , versus the conventional shape factor
d/a for various mobility ratios. The analytical solution for the unit mobility ratio (Morel-
Seytoux, 1966) is also plotted on the same figure for comparison. We find a good match
of the sweep efficiency at breakthrough between the analytical solution and simulation
results.
For unit mobility ratio, Fig. 4 teaches that a staggered-line-drive pattern always
has better areal sweep efficiency than a five-spot pattern. As the staggered-line-drive
pattern becomes longer relative to its width, the displacement pattern approaches linear
flow. High sweep efficiency results.
As the mobility ratio becomes more favorable, the advantage of staggered line
drive on sweep efficiency diminishes. When the mobility ratio decreases to 0.2, the five-
spot pattern becomes better than the staggered-line-drive pattern with d / a = 1. However,
if d / a is increased, the staggered line drive recovery is better than the five spot pattern
for this mobility ratio.
When the mobility ratio decreases to 0.05 or lower (very favorable), the areal
sweep efficiency for the five-spot pattern is very close to 1 at breakthrough. That is,
sweep out is complete at breakthrough. At this mobility ratio, the five spot pattern is as
good as a very long staggered line drive (d / a ≈ 15, almost linear flow), and much
better than the common staggered line drive (d / a = 1). The transition point for five
spot sweep efficiency exceeding that from a staggered line drive is around a mobility
ratio of 0.3. That is to say, if the mobility ratio is higher than 0.3, a staggered line drive
is always better than a five spot. If mobility ratio is lower than 0.3, then the five spot can
be higher in sweep efficiency than a staggered line drive.
11
Fig 4 Effect of Mobility Ratios and Patterns On Areal Sweep Efficiency
19
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12
The excellent displacement from a five spot pattern for very favorable mobility
ratios is explained with the help of streamline distributions. Every pair of streamlines
forms a stream tube, and the volumetric flow rate is the same in all the stream tubes. All
of the stream tubes connect with the same injector and producer, and the pressure drop
for all the streamtubes is the same. With the same pressure drop and the same volumetric
flow rate, the flow resistance is the same for all the streamtubes.
For our straight-line relative permeability assumption, we have
Ri = −∆p / q =µkA
dl0
Li∫ (7)
where Ri and Li are the resistance and length of streamtube i , respectively, k is the
homogeneous permeability, and A is the cross sectional area of the streamtube. The
cross-sectional area is
A = hw
where h is the constant thickness of the layer and w is the width of the streamtube.
Resistance in streamtube i is the same as that in streamtube j and thus
µkA
dl0
Li∫ = C (8)
for all the streamtubes at a given time, where C is constant. Moving the constant
parameters k and h to the right hand side, we have
µw
dl0
Li∫ = C (9)
For a piston-like displacing front,
µ µ1
0
2
1 1
wdl
wdl C
l
l
Lfi
fi
i
∫ ∫+ = (10)
13
where µ1 and µ2 are the constant viscosity of the displacing and displaced fluids,
respectively, and l fi is the distance from the injector to the displacing front.
1.5.2 Unit Mobility Ratio
For a unit mobility ratio, the pressure field remains unchanged throughout the
displacement, and so do the streamlines. The streamline distributions at M = 1 for the
five-spot and staggered-line-drive (d / a = 1) patterns are shown in Fig. 5.
For unit mobility ratio, Eq. (10) is
µ 1
w0
Li∫ dl = C (11)
From Eq. 11, we know that if the ith streamtube is longer than the jth streamtube, then the
average width of the ith streamtube wi is greater to maintain the same resistance and
flow rate. Therefore, the volume of the ith streamtube is larger than the jth streamtube.
The greater the difference in streamtube length, the bigger the difference in streamtube
width, and, when breakthrough happens in the jth streamtube, the front has not
progressed as far in the ith streamtube.
For a five-spot pattern, the longest streamline is that along the boundary, which is
2a. The shortest streamline is the one along the diagonal, at a length of 2 a . The ratio of
the longest streamtube length over the shortest is 2 . Since the width of the longest
streamtube is also 2 times as great as the shortest streamtube, the volume of the
longest streamtube is twice that of the shortest.
However, for a staggered-line-drive pattern with d/a = 1, the ratio of the length
along the boundary (3a/2) over the diagonal( 5a / 2)is 3 / 5 ≈ 1.34 . From the
streamline distribution in Fig. 5, we know that the shortest streamline is longer than
diagonal, and therefore the ratio of the longest streamline over the shortest is less than
1.34. This ratio is about 1.3 and, therefore, less than that ratio for a five spot pattern
which is 1.41. That is to say, the streamlines are more evenly distributed in the staggered
14
Fig 5 Saturation and Streamline Distributions, Unit Mobility Ratio
(a) Five Spot Pattern
(b) Staggered Line Drive
tD = 0.204
tD = 0.408
tD = 0.612
tD = 0.68
tD = 0.714
tD = 0.748
tD = 1.02
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tD = 0.208
tD = 0.416
tD = 0.624
tD = 0.763
tD = 0.783
tD = 0.797
tD = 0.832
tD = 1.04
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15
line-drive than in a five-spot pattern. Therefore, when the shortest streamtube breaks
through, a larger portion of the other streamtubes have been swept in a staggered-line-
drive than in a five-spot pattern. When d / a increases, the streamtube length ratio
(longest to shortest) decreases. When breakthrough happens in the shortest streamtube, a
greater portion is swept in the longest streamtube resulting in higher sweep efficiency at
breakthrough.
1.5.3 Five-Spot Pattern, Very Favorable Mobility Ratio
For a favorable mobility ratio ( M < 1), the displacement is stable. For equal
volumetric flow rate streamtubes, Eq. 10 holds. Here, we consider the case of very
favorable mobility ratio, i.e., the displacing fluid viscosity is much higher than that of the
displaced fluid. When the front moves a portion of the way down the streamtube, the
pressure drop is mainly in the displacing phase. That is to say, after a short injection time
(compared to breakthrough), the pressure drop in the displaced phase is negligible.
Therefore, Eq. 10 can be simplified to the following form
1
w0
lti∫ dl ≅ C (12)
In this extreme case, the displacing phase does not feel the producer until it is very close
to it because the pressure drop between the front and the well plays a negligible role in
the displacement. Initially, flow around the injector is radial, because the pattern appears
to be infinite at short times. For example, examine Fig. 6b for tD = 0.583. However, after
the front reaches a corner of the pattern, the no-flow boundary condition along pattern
borders alters the radial flow pattern. Pressure isobars must intersect the no-flow
boundaries at 90o. This constrains the streamlines in the region adjacent to a boundary to
be parallel to it. Because the fluids are incompressible, streamlines cannot terminate. The
flow field in the region near the front transitions from radial to quasi-radial. From the
figure, we also see that streamtubes ahead of the front are narrower along the boundary
than those in the center, which makes the front in the boundary streamtubes move faster
16
Fig 6 Five Spot Pattern, Very Favorable Mobility Ratio(1/M=20)
(a) Saturation Distribution
(b) Streamline Distribution
tD = 0.583
tD = 0.796
tD = 0.837
tD = 0.878
tD = 0.918
tD = 0.938
tD = 0.969
tD = 0.998
tD = 0.583
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17
than in the central streamtubes. Little area is unswept and the sweep efficiency at
breakthrough approaches unity.
Similar to before, when the path of displacing fluid in the ith streamtube is longer
than that in the jth streamtube, then the ith streamtube is wider in the swept region to
keep the same resistance (same flow rate and pressure). Therefore, the streamtubes along
the boundary become wider near the corner where the streamtube changes direction.
Streamlines remain smooth. This streamline distribution trend is apparent in Fig. 6b.
In summary, the very favorable mobility ratio conspires with boundary conditions
to determine the way that streamlines evolve, and makes the sweep efficiency at
breakthrough near unity. If the mobility ratio is very favorable, the pressure drop is
mainly in the displacing phase, and it does not feel the well, but is affected by the
boundary.
1.5.4 Staggered Line Drive, Very Favorable Mobility Ratio
For the staggered line drive pattern, the displacement at the beginning is similar to
that in the five-spot pattern. That is, the displacement pattern is radial around the injector
before the front reaches the nearest corner. The differences in displacement behavior
between the two patterns occurs after the front reaches the near corner. For a five spot,
because of the symmetry, the front reaches the two corners at the same time. However,
for a staggered line drive, the front reaches the closest corner first.
After the front passes the near corner, the streamlines evolve in a way similar to
the five spot. The streamtubes along the boundary are wide near the corner but narrow
near the front. This makes the front near the boundary move faster because the
streamtubes are narrower than those in the center of the pattern. Therefore, the front near
the boundary on the near no-flow corner side catches up, and the displacement
approaches linear flow (see the relevant streamline distribution in Fig. 7 at tD = 0.60 and
0.80). If the aspect ratio is large, flow in the center of the pattern must become nearly
linear because the pressure isobars are nearly straight and intersect the pattern boundary
at 90o.
18
Fig 7 Staggered Line Drive, Very Favorable Mobility Ratio(1/M=20)
(a) Saturation Distribution
(b) Streamline Distribution
tD = 0.20
tD = 0.40
tD = 0.60
tD = 0.80
tD = 0.88
tD = 0.90
tD = 0.92
tD = 0.94
tD = 1.0
tD = 0.20
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19
Similar to the discussion above for a five-spot pattern, for a very favorable
mobility ratio, the displacing front is perpendicular to the borders of the pattern both
before and after the front passes the nearest corner. As a result, we see linear
displacement for some time until the front on the near-corner side approaches the
producer. We see these front shapes in Fig. 7.
When the front approaches the producer, the streamtubes narrow due to the
confinement of the pattern boundary and the well. And therefore, with the same flow rate,
the displacing fluid will break through relatively quickly in the streamtubes closest to the
producer. The front on the far no-flow side progresses more slowly. This streamline
distribution does not change greatly as the mobility ratio becomes more favorable.
Sweepout of the pattern is not complete at breakthrough. For instance, a small amount of
the resident fluid remains along the right hand boundary as shown in Fig 7a at tD = 0.94 .
If the length of staggered line drive is increased (increasingd / a ), then the
displacement will approach linear flow and the sweep efficiency will approach unity. The
proportion of unswept area decreases as d / a increases.
1.6 CONCLUSIONS
Pattern performance changes with mobility ratio. For unit mobility ratio,
unfavorable mobility ratios and some favorable mobility ratios (M > 0.3), a staggered-
line-drive pattern has higher areal sweep efficiency than a five-spot pattern. However, for
very favorable mobility ratios (M < 0.3), a five-spot pattern has better sweep efficiency
than a common staggered-line-drive.
The reason for this behavior is the change of streamline and pressure distributions
with mobility ratios. For very favorable mobility ratios, the displacing front is near an
isobar and intersects the pattern boundary at 90o. This causes the fronts at times near
breakthrough to become radial around the producer for a five-spot pattern. This
displacing front shape is due to the symmetry of the five spot pattern.
20
For a staggered line drive, the displacing front is also perpendicular to the border
of the pattern. However, because the pattern is not symmetric, sweepout at breakthrough
is not complete. Only in the limit of very large d/a will the areal sweep efficiency
approach 1.
The simulation results are quite close to the analytical solutions for unit mobility
ratio. The results are also very close to the experimental data, Dyes et al. (1954), after
breakthrough at various mobility ratios. We find physical dispersion in the Dyes et al.
experimental results that cause earlier breakthrough time.
We observed some numerical dispersion in our simulation results. For very
favorable mobility ratios, the dispersion is small. We corrected the simulation results by
fitting the fractional flow curve with a second order polynomial to estimate breakthrough
time.
21
Appendix A—Effect of Grid-Block and Time-Step Size
The number and size of grid blocks and time steps has a strong effect on
simulation results. The extent of the effect is related to the mobility ratio. This study is
performed on five-spot pattern cases and the results are shown in Fig. A. However, the
conclusions drawn from this study do not lose any generality.
A.1 EFFECT OF TIME-STEP SIZE
Figure A plots the fractional flow at the producer versus time; changes in this
curve versus the number of grid blocks or time step are a convenient measure of the
accuracy of a solution. Comparing the effect of time-step size between M=1 and M=1/20
in Fig A, it is obvious that the effect of time-step size is much stronger in the very
favorable mobility ratio case (M=1/20). For unit mobility ratio (M=1), simulation results
stop changing at 50 time steps. However, for M=1/20, the difference between simulation
results and the converged solution for the same number of time steps is much greater. The
reason is described below.
For unit mobility ratio, the pressure field as well as the streamline distribution
does not change with time. If we map the analytical solution along the streamline, then,
time-step size does not have any effect on the simulation result. However, I mapped the
numerical solution to be consistent with the very favorable mobility rstio cases in which
mapping the analytical solution is not applicable.
As mobility ratio departs from unity, the pressure field changes more intensively.
For a very favorable mobility ratio, we must solve the pressure field enough times to get
a converged solution. We can see that, for M=1/20, the simulation result converges at
100 pressure solves (time steps).
For the calculations presented in the main portion of the report, I used 100 time
steps for unit mobility ratio and 300 time steps for M=1/20. The simulation result is
converged in terms of number of time steps.
22
23
A.2 EFFECT OF GRID-BLOCK AND TIME-STEP SIZE
Figure A also shows that the effect of grid-block size is much stronger in the unit
mobility ratio cases than it is in the very favorable mobility ratio cases. For M=1/20, the
simulation results converge when the number of grid blocks is greater than 10 by 10.
However, for unit mobility ratio, results converge for much finer grid blocks (100 by
100).
In very favorable mobility ratio cases, the simulation result is not sensitive to the
grid-block size. The reason can be the very sharp displacing front for very favorable
displacement.
I used 100 by 100 grid blocks for all the cases in this study, and therefore, the
simulation results are converged in terms of grid-block size.
A.3 CONCLUSION
In the unit mobility ratio cases, simulation results are sensitive to the grid-block
size but not very sensitive to time-step size. In the very favorable mobility ratio cases,
the simulation results are sensitive to grid-block size. For the calculations presented in
the main portion of the report, I used enough number of time steps and grid blocks for a
converged simulation result. Therefore, the results above demonstrate that simulations
are free from effects of the size of grid blocks and time steps.
24
Appendix B—Effect of Orientation of Grid-Blocks
We thought that the orientation of the grid-block with respect to the flow direction
may affect the simulation results, especially the streamline distribution. Therefore, I
studied this effect. In the work presented before, the injector and producer are aligned
with the diagonal as shown in Fig. B1a. It was not clear whether the streamline
distribution changes if streamlines align with the grid block (see Fig. B1b). Again, I only
studied the five-spot pattern. However, the conclusion is general for other patterns.
(a) (b)
Fig. B1 Grid Orientation.
I ran the simulation for both cases illustrated in Fig. B1 with time steps and grid-
block size remaining the same.
In 3DSL only rectangular grid blocks can be used, in the rotated grids. Therefore,
I have a bigger domain with the quarter of five-spot pattern sitting in the middle. A very
low permeability was assigned to regions outside the five-spot domain. In doing so, there
is essentially no flow outside the five-spot domain. The no-flow boundary is
approximated by zigzag lines along the boundary. However, as I used many grid
blocks(100 by 100), the boundary is quite smooth.
Because the grid block in the no-flow corner is closed on three sides, it is not
open to flow and it is not filled with the displacing phase when the surrounding blocks in
the five-spot pattern domain are swept by the displacing phase (see Fig. B2). It causes
the breakthrough time to be slightly earlier (about 0.1%). However, it does not make
much difference to the overall pattern behavior.
25
There is also some dispersion into the region outside the five-spot domain, as
illustrated in Fig. B2, due to the nonzero permeability in that region. 3DSL does not
work for zero permeability. This dispersion is small and does not affect the pattern
behavior.
Comparing Fig. B2 with Fig. 6, we can easily see that the direction of the grid
does not effect the pattern behavior. Therefore, any difference in the pattern behavior at
different mobility ratios is not due to improper grid orientation.
26
27
Appendix C—Numerical Dispersion
Fig. C shows the position of the displacing front at times near breakthrough (tD =
0.95) for the case of a five-spot pattern and a very favorable mobility ratio, M=1/20. The
solid line plots front location in the streamline closest to the boundary, whereas the
dashed line corresponds the diagonal line between injector and producer. The purpose of
this plot is to study the numerical dispersion for this case, because we think that the
breakthrough time should be unity in the absence of numerical dispersion.
I assumed piston-like displacement in choosing relative permeability curves.
However, because I mapped the numerical solution along the streamlines, this sharp front
may be smeared by numerical dispersion. However, due to the numerical-sharpening
effect for very favorable mobility ratios, the front is still quite sharp.
From Fig. C, it is obvious that the displacing front is very sharp. However, we
can still clearly see that it is not exactly a piston-like displacement due to the numerical
dispersion. The dispersion is very small. If there is no dispersion, the piston-like
displacing front bisects the numerically predicted front. The dimensionless width of the
front is 0.02, and the perfect piston-like front is in the middle. A dimensionless distance
of 0.01 from the front in the simulation result. Therefore, the numerical dispersion
causes 0.01 (tD) earlier breakthrough in the earliest breakthrough streamline. Referring to
Fig. 2(a), the calculated breakthrough tD for this case is over 0.99. Therefore, without
numerical dispersion, there will be perfect displacement for the five-spot pattern at very
favorable mobility ratio.
Fig. C also tells us the front shape along the boundary is almost the same as that
along the diagonal. Therefore, the slanted front is only due to the numerical dispersion,
not the boundary effect.
28
Fig. C Displacing FrontFive Spot Pattern, 1/M = 20
30
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29
2 STREAMLINES TO SOLVE INVERSE PROBLEMS
2.1 DEFINITION OF THE RESEARCH TOPIC
The proposed research is to apply the concept of streamlines to infer permeability
fields based on the tracer breakthrough curve and pressure differences between the
injector and producers. The basic idea of my approach is to adjust the streamlines to
match the reference fractional flow and pressure data, and through streamlines to modify
the underlying permeability field.
Because there is no flow across a streamline, we can represent the flow field by
1D flow streams along the streamlines. Each streamline is associated with a time of flight
which indicates the breakthrough time. In the 3DSL streamline simulator (Batycky et al),
we solve the pressure field at given saturation distribution, obtain the streamline
distribution, then map the Buckley-Leverett solution along the 1D streamlines for a short
time period, and then solve the pressure field again and repeat the process. Because the
pressure equation and saturation equation are decoupled in 3DSL, the simulation is
speeded up significantly.
Conventional reservoir simulation history matching is time consuming because of
the large number of grid blocks. However, if we know the time of flight of the
streamline, then we will know the breakthrough time of an individual streamline.
Therefore, if the breakthrough curve for our computed permeability field does not match
the reference breakthrough curve, then we know along which streamline and in which
way to modify the permeability for a better match.
This approach is described below. To start with this project, I made the
following simplifications:
• Incompressible flow, because the streamline simulator (3DSL by Batycky et al) that I
am using works for incompressible fluid flow only;
• Piston-like displacement;
• Unit mobility ratio, valid for tracer flow study;
30
• Two dimensional;
• No constraint of the permeability value or its distribution.
Steps of the Streamline Approach to the Inverse Problem:
1. Given a reference permeability field, use the 3DSL streamline simulator for forward
simulation to obtain the reference water breakthrough curves at producers and
pressure drops between injector and producers.
2. Start from an initial uniform permeability field. Do the same simulation. Check the
match—both breakthrough and pressure drop. If it does not match the reference data,
modify the permeability as in the following steps;
3. Work on the streamlines: Calculate the time of flight for all streamlines using 3DSL
to output the coordinates of the streamlines. Sort the streamlines in the order of time
flight.
4. Compare the computed and the reference breakthrough and pressure drops. For the
breakthrough curve, check where in the curve the difference lies. Relate it to the
corresponding streamline, and then change the permeability value depending on the
difference of the breakthrough curves;
5. Repeat the simulation using the new permeability distribution. This completes one
iteration;
6. Repeat steps 2 to 5 until the match is good enough.
I obtained very promising results with this approach. In the reference field, there
are one injector and two producers. Between the injector and producer there exists a high
permeability channel, and between the injector and producer 2 there is a low permeability
barrier (See Fig. 1). Starting from a homogeneous permeability field, both the
breakthrough and pressure drops are far away from the reference (see Fig. 2). After three
iterations, I obtained a very good match both in pressure drops and breakthrough curves
(Fig. 2), which indicates that this approach converges very fast, much faster than most of
other approaches to this inverse problem.
31
Fig. 1 Comparison on Permeability Field
Producer 1
Producer 2
36
Reference Permeability Field
Computed Permeability Field after 3rd Modification
32
Fig. 2 Comparison on Breakthrough Curve
37
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33
Fig. 3 plots the error versus the number of iterations. The error is defined as
[ ]E E Efw i p ii
n prod
= +=∑ , ,
_
∆1
(1)
where Efw is the average absolute error in the fractional flow at different times, and E p i∆ ,
is the relative error of the pressure drops between the injector and producers at producer i.
They are defined below
En
f ffw i w i jC
w i jR
j
n
, , , , ,= −=
∑1
1
(2)
where fw i jC, , and fw i j
R, , are the computed and reference water fractional flow during time
step j at producer i, respectively.
Ep p
pp iiC
iR
iR∆
∆ ∆∆, = −
(3)
where ∆piC and ∆pi
R are calculated and reference pressure drop between the injector and
producer i, respectively.
I normalize the error in pressure by the reference pressure so that it will be within
the range of 0 to 1 and it is of the same scale as the error in fractional flow. Then I can
sum up them up to have a unified error estimation.
The solid line in Fig. 3 (a) is the sum of error at the two producers. The dashed
and dotted line are for each producer, respectively.
Comparing the computed permeability field with the reference field in Fig. 1, we
find that, with this approach, the high permeability channel is retrieved quite accurately.
For the barrier, the resolution is not very good. However, we have good matches for the
breakthrough curve and pressure drop.
34
(b)
Fig. 3 Error vs Iterations
38
0.0
0.2
0.4
0.6
0.8
1.0
Err
or
0 1 2 3(a)
a
a
a
a
b
b
b
b
a
a
a
a
a
b
a
TotalProducer 1Producer 2
0.0
0.2
0.4
0.6
Err
or
0 1 2 3Iteration
a
a
a
a
b
b
b
b
a
a a
a
b
b
b
b
a
b
a
b
E∆p at Producer 1Efw at Producer 1E∆p at Producer 2Efw at Producer 2
35
2.2 IMPORTANCE OF THE RESEARCH
History matching plays an important role in reservoir engineering. It is important
for prediction and data interpretation. In many cases, we have water or tracer
breakthrough and pressure information at the producers. This information contains much
information about the permeability distribution of a reservoir, but it is difficult to infer
this distribution. There are many approaches to this inverse problem. Most approaches
manipulate parameters at the grid-block level that corresponds to conventional simulation
grids. Because there are many cells, the optimization problem is large. By conducting the
optimization at the streamline level, the opportunity for speed improvement is great.
Since we know how each streamline affects water cut, we can speed up the inverse
process tremendously. A more typical approach would be to perturb parameters in a
more random fashion to gauge the effect of the parameter.
2.3 FUTURE WORK PLAN AND EXPECTED RESULTS
There are many simplifications in my current study. Simplification means
limitations in the application. Therefore, my future work will be focused on relaxing the
simplifications to broaden its application. My plan is to
• Release the piston-like displacement and unit mobility ratio assumptions. It is not
difficult to do since I can simply map the Buckley-Leverett solution instead of the
sharp front, and I can map the numerical solution for one-dimensional displacement
instead of the analytical one along the streamline.
• Seek a streamline simulator for compressible flow. Study my current method for
modifying a permeability field to see whether it is still valid for compressible flow.
Also study what modification of the current approach should be made for
compressible flow. Then I can make this approach work for primary recovery and
well-test problems. This will be the most difficult portion and therefore the key part
of this project.
36
• Put constraints on the permeability distribution from other information such as
seismic data, and broaden its use to three-dimensional problems. Improve the
method.
• For comparison, develop examples that use a conventional history-matching approach
to infer heterogeneity.
I will continue to work on this topic in my Ph.D. When this project is finished, I
expect that this approach will work for very general history matching purposes.
37
NOMENCLATURE
A area, L2
AS area swept, L2
AT total area of the pattern, L2
a distance between like wells (injection or production) in a row, L
d distance between adjacent rows of injection and production wells, L
EA areal sweep efficiency
fw fractional flow of water
h bed thickness, L
k permeability, L2
kro relative permeability of oil
krw relative permeability of water
l length, L
Li length of stream tube i, L
lfi distance of the displacing front from the injector in stream tube i, L
M mobility ratio
p pressure, M/L • T2
q flow rate, L3/T
Ri flow resistance of stream tube i, M/L4 • T
S saturation
Sw water saturation
tD dimensionless time
VI volume of displacing phase injected, L3
w width of a stream tube, L
φ porosity
µ viscosity, M/L• T
38
REFERENCES
1. Batycky, R. P., Blunt, M. J., and Thiele, M. R.: “A 3D Streamline-Based ReservoirSimulator” SPE Reservoir Engineering, 12, 246, (1997).
2. Craig, F. F. Jr.: “The Reservoir Engineering Aspect of Water Flooding,” Society ofPetroleum Engineers Monograph, Dallas, TX, 1971.Dyes, A. B., Caudle, B. H., andErickson, R. A., “Oil Production after Breakthrough—as Influenced by MobilityRatio”, Petroleum Transactions, AIME, 201, 27 (1954).
3. Dyes, A.B., Caudle, B.H., and Erickson, R.A.: “Oil Production AfterBreakthrough—as Influenced by Mobility Ratio,” Petroleum Transactions, AIME,201, 27 (1954).
4. Lake, L. W.: Enhanced Oil Recovery. Prentice Hall Inc., New Jersey, 1989.
5. Morel-Seytoux, Hubert J.: “Unit Mobility Ratio Displacement Calculations forPattern Floods in Homogeneous Medium,” SPE J, 6, 217, (1966).
6. Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation. ElsevierScientific Publishing Co., New York, 1977.
7. Thiele, M. R., Batycky, R. P., Blunt, M. J. and Orr Jr, F. M. Jr.: “Simulating Flow inHeterogeneous Systems Using Streamtubes and Streamlines,” SPE. Reser. Eng, 11, 5,(1996).
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